Counted cross stitch should be done with a tapestry needle. Tapestry needles have blunt points and much larger eyes than sewing needles. The blunt points prevent the needles from piercing fabric threads.
Tapestry needles come in a variety of sizes. A larger size number means a smaller needle. Cross stitching usually requires a #22, #24, #26 or #28 needle.
One traditional rule says you should use a #22 needle if the fabric is 14 count (14 threads per inch) or less, a #24 or #26 needle if the fabric count is 16-18 count, and a #26 needle if the fabric is finer than 18.
The needle should be large enough to move the fabric threads out of the way just a tiny bit. This reduces the friction and wear on your stitching fiber.
The floss or fiber thickness and number of strands used can also affect the choice needle size.
The usual "rule" holds--find a size (or sizes) you like.
Some people lose the finish on their needles over time. Besides being ugly, this can make the needle more difficult to use. Special finishes, such as gold and platinum, are available. They cost more but some stitchers find they last longer. Try different finishes until you find the one that works best for you.
Chair arms are very convenient for holding needles, but such use can cause other members of the household to acquire a more intimate acquaintance with the tools of your craft than either they or you desire. A pin cushion is an obvious solution. Needle safes also work well. These are small, flat cases lined on the inside faces with magnets. Needle safes can cost from US$5 for a small plastic one to more than US$30 for a good, handcrafted, wood-and-brass box. People have also had good results with magnetic paperclip holders, available in any place that sells office supplies.